In a slightly less general PTQM
setting, the key role is
still played by the
details of the metric-mediated transition from the auxiliary
Hilbert space
\({\cal H}_{math}\)  to its correct physical partner
\({\cal H}_{phys}\) . In a way indicated by the horizontal arrow
in diagram (REF ) the fundamental
message that
the Hamiltonian \(H\)  must be Hermitian in \({\cal H}_{phys}\) 
remains unchanged.
In the literature, unfortunately, this message is often obscured by
the diversity of notation conventions
(see their sample in Table Nr. 1 of [1]}).
We will use here, therefore,
the properly modified Dirac's
bra-ket formalism [2]}
and the maximally
compact abbreviations
for \({\cal H}_{phys}= {\cal R}_{0}\) 
and
\({\cal H}_{math}= {\cal R}_{N_{}}\) .
This will enable us to distinguish easily
between the quantum mechanics of textbooks
(in which we may put \(N_{}=0\)  to imply that
\({\cal H}_{math}\ \equiv \ {\cal H}_{phys}\ \equiv \ {\cal L}_{}\) )
and the menu of the two-space CHSP scenarios
in which one may choose \(N_{}\ge 1\) 
emphasizing
the non-equivalence between
\({\cal H}_{math}= {\cal R}_{N_{}}\) 
and \({\cal H}_{phys}= {\cal R}_{0}\) .
